A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

Assyr Abdulle (1) , Gilles Vilmart (2, 3)

1 EPFL - Ecole Polytechnique Fédérale de Lausanne
2 IPSO - Invariant Preserving SOlvers
3 IRMAR - Institut de Recherche Mathématique de Rennes

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L^2 and the H^1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.

Mots clés: nonmonotone quasilinear elliptic problem; a priori error estimates; numerical quadrature; variational crime; finite elements

Domaines: Analyse numérique [math.NA]

Assyr Abdulle, Gilles Vilmart. A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems. Numerische Mathematik, 2012, 121, pp.397-431. ⟨10.1007/s00211-011-0438-4⟩. ⟨hal-00746826⟩